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Centric and acentric reflections: Difference between revisions
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Centric reflections play the same role in reciprocal space as special positions in real space: they may occur at the borders of the asymmetric unit. It makes sense to discuss the reciprocal case and the real space together. | Centric reflections play the same role in reciprocal space as special positions in real space: they may occur at the borders of the asymmetric unit. It makes sense to discuss the reciprocal case and the real space together. | ||
A definition and a theorem about centric reflections are stated here before the role of centrics is examined . | A definition and a theorem about centric reflections are stated here [1] before the role of centrics is examined. | ||
Definition: '''A reflection (h,k,l) is said to be centric if in the space group there is at least one symmetry operation g(x)=R_g*x+t_g whose rotational part R_g sends the reflection to minus itself''', i.e.: | Definition: '''A reflection (h,k,l) is said to be centric if in the space group there is at least one symmetry operation g(x)=R_g*x+t_g whose rotational part R_g sends the reflection to minus itself''', i.e.: | ||
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(h,k,l) is centric if there is a symop g(x)=R_g*x+t_g in G such that R_g*(h,k,l)=(-h,-k,-l) | (h,k,l) is centric if there is a symop g(x)=R_g*x+t_g in G such that R_g*(h,k,l)=(-h,-k,-l) | ||
Theorem: '''The phase of a centric reflection is restricted to phi(h,k,l)= | Theorem: '''The phase of a centric reflection is restricted to phi(h,k,l)=pi*(h*tx_g+k*ty_g+l*tz_g) plus or minus any integer number of pi''' | ||
where the vector t_g=(tx_g,ty_g,tz_g) is the translational part of the symop g that causes the reflection to be centric. | where the vector t_g=(tx_g,ty_g,tz_g) is the translational part of the symop g that causes the reflection to be centric. | ||